Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 315754, 8 pages
doi:10.1155/2010/315754
Research Article
Abnormal Curves on the Goursat Systems of Rn
Mohamad H. Cheaito and Hassan Zeineddine
Department of Mathematics, Faculty of Sciences, Lebanese University, Hadath, Lebanon
Correspondence should be addressed to Mohamad H. Cheaito, mohamad cheaito@yahoo.com
Received 9 May 2010; Accepted 27 December 2010
Academic Editor: Frédéric Robert
Copyright q 2010 M. H. Cheaito and H. Zeineddine. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
We study the abnormal and the rigid curves of the 2-distributions of Rn satisfying everywhere the
Goursat condition. We give the directions for the rigid and the abnormal curves when the systems
satisfy the strong Goursat condition or when they have a singularity of order 2 in each dimension.
1. Introduction
Let E be a 2-distribution on Rn . We denote by
E1 E1 E,
Ei Ei−1 , Ei−1 ,
Ei E, Ei−1 .
1.1
A small growth vector sgv of E, at a point p ∈ Rn , is the sequence
r1 p , r2 p , . . . S ,
1.2
where ri p dim Ei p, for every i ≥ 1.
The great growth vector, at p, is the sequence
m1 p , m 2 p , . . . G ,
1.3
where mj p dim Ej p, for every j ≥ 1.
If the dimensions of Ei resp., Ej are independent of p, then the distribution is called
regular resp., totally regular.
If the great growth vector, at a point p ∈ Rn , is 2, 3, 4, . . . , nG , then the distribution
is called distribution satisfying the Goursat condition at p. Moreover, if E satisfies, on a
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neighborhood of p, the Goursat condition, then its annihilator, E⊥ , is called Goursat system
and denoted by GS.
The classification of the distributions, with respect to the small and great growth
vectors, was the object of many articles. The beginning was by Engel 1, where he gave
the normal form of the GS in dimension 4.
In an article written in 1910, Cartan 2 studied the case of dimension 5. In 1978 Giaro
et al. completed the work of Cartan about the systems of dimension 5 3. In such a case
2 nonequivalent models are presented. In 1981, Kumpera and Ruiz 4 gave the different
normal forms in dimension n ≤ 6.
The classification, of models, in dimensions 7 and 8 are given by 5. The study of the
models in dimension n is also open. We say that 6, when the small and the great growth
vector are the same, we have the system GNF.
Zhitomirskiı̆ 7 gave the asymptotic normal forms of the regular distributions and the
generic case studied in many articles, for example 8.
The normal form of the model, satisfying at a neighborhood of a point the small growth
vector 2, 3, 4, 4, 5, 5, . . . , n − 1, n − 1, nS , is given in 9.
2. Rigid and Abnormal Line Subdistributions of the Goursat Systems
Satisfying the Strong Condition of Goursat
The Goursat systems are given by the following theorem.
Theorem 2.1 see 4, 5. Let E be a 2-distribution on Rn , satisfying in each point, the condition of
Goursat, then
E⊥ ⎧
⎪
ω1 dx2 x3 dx1 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ω2 dx3 x4 dx1 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎨ω3 dxi3 x5 dxj3 ,
⎪
ω4
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ω
dxi4 X6 dxj4 ,
n−2
i3 , j3 ∈ {4, 1, 1, 4},
i4 , j4 ∈ 5, j3 , j3 , 5 ,
2.1
..
.
dxin−2 Xn dxjn−2 ,
in−2 , jn−2 ∈ n − 1, jn−3 , jn−3 , n − 1 ,
where
Xl ⎧
⎨xl ,
⎩x c ,
l
l
if il−2 , jl−2 jl−3 , l − 1 ,
if il−2 , jl−2 l − 1, jl−3 ,
for 6 ≤ l ≤ n and c6 , c7 , . . . , cn−2 are real arbitrary constants.
This theorem gives the different Goursat systems denoted by GS.
2.2
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3
Definition 2.2. Let E be a 2-distribution on Rn , p ∈ Rn . E satisfies the strong condition of
Goursat, at p, if the small and the big growth vectors, at this point, are 2, 3, . . . , nS and
2, 3, . . . , nB .
Theorem 2.3 see 6. Let E be a 2-distribution on Rn satisfying, in each point, the condition of
Goursat. Suppose that E satisfies the strong condition of Goursat, at a point p ∈ Rn , then there exists
a local coordinate system x, U, around p, such that
E⊥ ⎧
⎪
⎪
⎪ω1 dx2 x3 dx1 ,
⎪
⎪
⎪
⎪
⎪
ω2 dx3 x4 dx1 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ω3 dx4 x5 dx1 ,
⎪
ω4
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ω
dx5 x6 dx1 ,
2.3
..
.
n−2
dxn−1 xn dx1 ,
it means that E is spanned by v1 ∂/∂xn and
v2 ∂
∂
∂
∂
− x3
− x4
− · · · − xn
.
∂x1
∂x2
∂x3
∂xn−1
2.4
Remark that, in this theorem, E satisfies the strong condition of Goursat, at a point p.
Such property can be extended without difficulty to a neighborhood of p. For the definitions
of abnormal and rigid curves, see 10.
Definition 2.4. Let E be a 2-distribution on M; a C1 -curve γ : α, β → M is said to be
horizontal or E-curve if γ . t ∈ Eγt , for any t ∈ α, β.
The set of horizontal curves connecting two points a and b of M, will be denoted by
φ, for any a, b ∈ M.
Ωa,b α, β. The theorem of Chow 11 certified that Ωa,b α, β /
Definition 2.5. Let E be a 2-distribution on M, a C1 -curve γ : α, β → M is said to be rigid, if
γ is an isolated point of Ωa,b α, β for the C1 -topology.
Definition 2.6. Let E be a 2-distribution on M. A line subdistribution i.e., distribution of
dimension one of L is said to be rigid, if any L-curve is rigid. L is said to be local rigid,
if for any p ∈ M, there exists a neighborhood U of p, such that any LU -curve is rigid.
If E is a 2-distribution on M, we denote Ωa α, β the set of E-curves γ : α, β → M,
starting from the point a.
Definition 2.7. A curve γ ∈ Ωa α, β, is said to be abnormal, if the mapping end:
Ωa α, β → M, defined by end γ γβ, is not a submersion at γ.
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Proposition 2.8 see 10. Let E be a k-distribution on M. If v1 , v2 , . . . , vk form a basis of E and
if γ ∈ Ωa α, β, such that γ . t u1 tv1 · · · uk tvk |γt , then the following propositions are
equivalent.
1 γ is abnormal.
2 There exists a lift curve Γ : α, β → T M, absolutely continuous, of coordinates
(q1 , q2 , . . . , qn ), such that
a Γt / 0, for any t ∈ α, β,
b Γt ∈ E⊥ ,
c Γ satisfies the equation q1 . , q2 . , . . . , qn . = u1 tq1 , q2 , . . . , qn dv1 · · · uk tq1 , q2 , . . . , qn dvk |γt .
Definition 2.9. Let E be a 2-distribution on M; a line subdistribution L is said to be abnormal,
if any L-curve is abnormal. L is said to be local abnormal, if for any p ∈ M, there exists a
neighborhood U of p, such that any LU -curve is abnormal.
Definition 2.10. Let E be a 2-distribution on M; a distribution D on M is said to be nice
with respect to E if D is an involutive distribution of codimension 2 such that Ep ⊂
/ Dp and
2
dimEp ∩ Dp 2, for any point p ∈ M.
Proposition 2.11 see 10. Let E be a 2-distribution on Rn and L be a line subdistribution on E.
Consider the following properties.
a L is locally rigid.
b L is locally abnormal.
c Locally L is the intersection of E and a nice distribution.
d dimadL∞ p < n, for every p ∈ Rn .
Then, one has the following implication:
(a)
2.5
(c)
(b)
(d)
Zhitomirskiı̆, in 10, conjectured that d ⇒ b, and he proved that a, b, c, and
d are not equivalent in general. Now we prove that, The properties are equivalent if the
distribution satisfies the strong condition of Goursat.
Theorem 2.12. Let E be a 2-distribution on Rn , n ≥ 4, satisfying in each point the strong condition
of Goursat, then the properties (a), (b), (c), and (d) are equivalent.
Proof. By Theorem 2.3, E is spanned, on a neighborhood U, by
v1 ∂
,
∂xn
v2 ∂
∂
∂
∂
− x3
− x4
− · · · − xn
.
∂x1
∂x2
∂x3
∂xn−1
2.6
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5
Let L be a line subdistribution satisfying d, and let u av1 bv2 be a generator of L.
We have
v2 , v1 ∂
,
∂xn−1
v2 , v2 , v1 ∂
.
∂xn−2
2.7
Easily, by induction, we say that
advi 2 , v1
∂
,
∂xn−i
∂
∂
i
v1 , adv2 , v1 ,
0,
∂xn ∂xn−i
2.8
for every i 1, 2, . . . , n − 2.
A simple induction shows that
adui v1 αi1 v1 αi2 v2
i−1
j
αij adv2 v1 bi advi 2 v1 ,
2.9
j1
where αj i , for j 1, 2, . . . , i, are C∞ functions on U to R. Because dimadL∞ p < n, for every
p ∈ Rn , we have necessarly b 0 and by consequently L is spanned by v1 .
Prove now d ⇒ c. Let Z kerdx1 ∧ dx2 , we say easily Z is a nice distribution
see 10. In fact: v1 x1 v1 x2 0, then v1 ∈ E ∩ Z. Otherwise v1 , v2 −∂/∂xn−1 , then
v1 , v2 x1 v1 , v2 x2 0, we deduce that E2 ∩ Z span{v1 , v1 , v2 } and consequently
dimE2 ∩ Zp 2.
Now codZ 2 and Z is integrable. Because v2 x1 0, we obtain Ep is not a subset
of Zp , for every p ∈ Rn , then Z is a nice distribution. Moreover L E ∩ Z, then d ⇒ c, by
10.
Prove now d ⇒ a. Consider the form ωn−2 of the system E⊥ . We have ωn−2 0 0 and
dxn−2 0 /
iv1 dωn−2 iv1 dxn−1 ∧ dx1 i∂/∂xn dxn−1 ∧ dx1 0.
2.10
0 and ω0 is not in E3 |0 . By
Otherwise v2 , v1 , v2 −∂/∂xn−2 then ωv2 , v1 , v2 −1 /
Theorem 5.7 of 10, L is locally rigid.
Let E be a 2-distribution of Rn , spanned by v1 and v2 . LE is the line subdistribution
spanned by a vector field in the form av1 bv2 , where a and b are such that av1 , v1 , v2 0. We say that LE is independent of the choice of v1 and
bv2 , v1 , v2 is in E2 and a2 b2 /
v2 . Zhitomirskiı̆ 10 proved that LE is a line subdistribution locally rigid, also by a conjecture,
it is unique, in the case where E is regular and satisfying the condition
dim E2 3,
this is the case of GS1 .
dim E3 4,
2.11
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3. Rigid and Abnormal Line Subdistributions of the Goursat Systems
Presenting in Each Dimension a Singularity of Order 2
Definition 3.1. Let S be a Goursat system. S is called presenting a transposition of order l,
l ∈ {3, 4, . . . , n − 2} if
ωl−1 dxil−1 Xl1 dxjl−1 ,
3.1
ωl dxjl−1 xl2 dxl1 .
Definition 3.2. If the small growth vector of a 2-distribution E on Rn , at a point p of Rn , has
the form 2, 3, . . . , s, s, . . . , s, . . . , n denoted by 2, 3, . . . , sk , . . . , n, the distribution is called
k times
a distribution presenting, in the dimension s, a singularity of order k.
Remark 3.3. If the distribution satisfies the condition of Goursat the dimensions 2, 3 and n are
of order 1 at every point.
Notation. The system of Goursat satisfying, at every point x
2, 3, 4k , 5k , . . . , n − 1k , nS is denoted by GSk .
∈
Rn , the condition
Theorem 3.4 see 9. Let E be a 2-distribution on Rn , satisfying at every point the Goursat
condition, such that at x0 ∈ Rn , we have 2, 3, 42 , 52 , . . . , n − 12 , nS . Then there exists a local
system of coordinates x, U, around x0 , such that
⎧
⎪
ω1 dx2 x3 dx1 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪ω2 dx3 x4 dx1 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
ω3 dx4 x5 dx1 ,
⎪
⎪
⎪
⎨
E⊥ ω4 dx5 x6 dx1 ,
⎪
⎪
⎪
⎪
..
⎪
⎪
.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
ωn−3 dxn−2 xn−1 dx1 ,
⎪
⎪
⎪
⎪
⎪
⎩
ωn−2 dx1 xn dxn−1 ,
3.2
it means that E is spanned by
v1 ∂
,
∂xn
v2 −xn
∂
∂
∂
∂
∂
xn x3
xn x4
· · · xn xn−1
.
∂x1
∂x2
∂x3
∂xn−2 ∂xn−1
3.3
Now we want to study the rigid and the abnormal line subdistributions directions
for the Goursat systems GS2 .
Definition 3.5. Let E be a 2-distribution spanned by v1 and v2 . The line subdistribution LE , is
the line subdistribution spanned by a vector field in the form av1 bv2 , where a and b are
0.
such that av1 , v1 , v2 bv2 , v1 , v2 ∈ E2 and a2 b2 /
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7
Theorem 3.6. In the Goursat systems (GS2 ), LE is the unique direction of abnormal and rigid curves.
Proof. E is spanned by
v1 ∂
,
∂xn
v2 −xn
∂
∂
∂
∂
∂
xn x3
xn x4
· · · xn xn−1
,
∂x1
∂x2
∂x3
∂xn−2 ∂xn−1
3.4
and E2 is spanned by v1 , v2 , and v1 , v2 , where v1 , v2 −∂/∂x1 x3 ∂/∂x2 x4 ∂/∂x3 · · · xn−1 ∂/∂xn−2 .
Prove now LE is spanned by v1 . In fact v1 , v1 , v2 0 and v2 , v1 , v2 ∂/∂xn−2 ,
then necessarily b 0 and LE span{v1 }. Recall that LE is a direction of rigid curves, then of
abnormal curves.
Does exist another direction field of the abnormal curves?
Let L Vect{αv1 βv2 } be an arbitrary line subdistribution of E. Let γ : I → Rn be a
horizontal curve of L, i.e., γ̇t ∈ Lγt . Suppose that γ is an abnormal curve. There exists a lift
curve Γ : I → T Rn satisfying the adjoint equation: ṗ1 , ṗ2 , . . . , ṗn −αp1 , p2 , . . . , pn dv1 −
βp1 , p2 , . . . , pn dv2 . In other hand:
⎛
0 0 0
⎜
⎜0
⎜
⎜
⎜0
⎜
⎜
⎜
ṗ1 , . . . , ṗn −β p1 , . . . , pn ⎜ ...
⎜
⎜
⎜0
⎜
⎜
⎜0
⎝
0 0 0 ···
0
0 ···
0
0 xn 0
0 0 xn 0 0
..
.
..
.
0 0
..
.
.. ..
. .
0 0 0
0 ··· ···
−1
⎞
⎟
x3 ⎟
⎟
⎟
· · · 0 x4 ⎟
⎟
⎟
..
.. ⎟.
.
. ⎟
⎟
⎟
0 · · · xn xn−1 ⎟
⎟
⎟
··· ··· 0 ⎟
⎠
0 0 ··· ···
··· ···
3.5
0
We verify that E2 Vect{v1 , v2 , ∂/∂xn−1 }. But Γ ⊂ E2 ⊥ , then we have
pn pn−1 0,
p1 x3 p2 x4 p3 · · · xn−1 pn−3 .
3.6
I
Suppose that β /
0. By the adjoint equation, we have ṗn−1 −βxn pn−2 0, but β /
0,
then pn−2 0. Similarly ṗn−2 −βxn pn−3 0, then pn−3 0.
Show that by induction pn−i 0, for every i 1, 2, . . . , n − 2.
For i 1, the property is true. Suppose that pn−i−1 0, prove that pn−i 0. By the
adjoint equation
ṗi −βxn pi−1 0
3.7
for every i 1, 2, . . . , n−3. we deduce that pi 0. Finally, using I we have p1 0. We deduce
Γ 0, impossible, then we obtain β 0 and L is spanned by v1 , by consequently L LE and
L is locally rigid.
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International Journal of Mathematics and Mathematical Sciences
Corollary 3.7. With the same conditions of Theorem 2.3, the distribution LE is the unique line
subdistribution locally rigid on E.
Proof. In fact, v1 , v1 , v2 0 and av1 , v1 , v2 bv2 , v1 , v2 bv2 , v1 , v2 −b∂/∂xn−2 ∈ E2 span{v1 , v2 , ∂/∂xn−1 } if b 0, then LE span{v1 }, but the distribution
spanned by v1 is the unique locally rigid subdistribution, on E, of dimension 1.
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